On univoque Pisot numbers. (English) Zbl 1182.11051
A real number \(\beta>1\) is said to be univoque if there is a unique sequence of integers \(s_1, s_2, s_3, \dots \in [0, \beta)\) such that \(1=\sum_{n \geq 1} s_n \beta^{-n}\). In this paper the authors study univoque Pisot numbers. In their main result they determine the smallest univoque Pisot number \(\beta=1.880000\dots\) which is the root of the polynomial \(x^{14}-2x^{13}+x^{11}-x^{10}-x^7+x^6-x^4+x^3-x+1.\) Its univoque expansion is \(111001011(1001010)^{\infty}\). The proofs are mainly computational. In particular, they use Boyd’s algorithm for finding Pisot numbers in an interval. It is also shown that the smallest limit point of the set of univoque Pisot numbers is the root \(\lambda = 1.905166\dots\) of \(x^4-x^3-2x^2+1\) which is itself a univoque Pisot number.
Reviewer: Artūras Dubickas (Vilnius)
MSC:
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11A67 | Other number representations |
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